^{*}

Conceived and designed the experiments: VMS. Analyzed the data: VMS EJD. Contributed reagents/materials/analysis tools: VMS EJD. Wrote the paper: VMS EJD WF.

The authors have declared that no competing interests exist.

Metabolic rate, heart rate, lifespan, and many other physiological properties vary with body mass in systematic and interrelated ways. Present empirical data suggest that these scaling relationships take the form of power laws with exponents that are simple multiples of one quarter. A compelling explanation of this observation was put forward a decade ago by West, Brown, and Enquist (WBE). Their framework elucidates the link between metabolic rate and body mass by focusing on the dynamics and structure of resource distribution networks—the cardiovascular system in the case of mammals. Within this framework the WBE model is based on eight assumptions from which it derives the well-known observed scaling exponent of 3/4. In this paper we clarify that this result only holds in the limit of infinite network size (body mass) and that the actual exponent predicted by the model depends on the sizes of the organisms being studied. Failure to clarify and to explore the nature of this approximation has led to debates about the WBE model that were at cross purposes. We compute analytical expressions for the finite-size corrections to the 3/4 exponent, resulting in a spectrum of scaling exponents as a function of absolute network size. When accounting for these corrections over a size range spanning the eight orders of magnitude observed in mammals, the WBE model predicts a scaling exponent of 0.81, seemingly at odds with data. We then proceed to study the sensitivity of the scaling exponent with respect to variations in several assumptions that underlie the WBE model, always in the context of finite-size corrections. Here too, the trends we derive from the model seem at odds with trends detectable in empirical data. Our work illustrates the utility of the WBE framework in reasoning about allometric scaling, while at the same time suggesting that the current canonical model may need amendments to bring its predictions fully in line with available datasets.

The rate at which an organism produces energy to live increases with body mass to the 3/4 power. Ten years ago West, Brown, and Enquist posited that this empirical relationship arises from the structure and dynamics of resource distribution networks such as the cardiovascular system. Using assumptions that capture physical and biological constraints, they defined a vascular network model that predicts a 3/4 scaling exponent. In our paper we clarify that this model generates the 3/4 exponent only in the limit of infinitely large organisms. Our calculations indicate that in the finite-size version of the model metabolic rate and body mass are not related by a pure power law, which we show is consistent with available data. We also show that this causes the model to produce scaling exponents significantly larger than the observed 3/4. We investigate how changes in certain assumptions about network structure affect the scaling exponent, leading us to identify discrepancies between available data and the predictions of the finite-size model. This suggests that the model, the data, or both, need reassessment. The challenge lies in pinpointing the physiological and evolutionary factors that constrain the shape of networks driving metabolic scaling.

Whole-organism metabolic rate, _{0} is a normalization constant and

Many other biological rates and times scale with simple multiples of 1/4. For example, cellular or mass-specific metabolic rates, heart and respiratory rates, and ontogenetic growth rates scale as ^{−1/4}, whereas blood circulation time, development time, and lifespan scale close to ^{1/4}

In a series of papers starting in 1997, West, Brown, and Enquist (WBE) published a model to account for the 3/4-power scaling of metabolic rate with body mass across species

Intensifying controversy has surrounded the WBE model since its original publication, even extending to a debate about the quality and analysis of the data

Much of the work aimed at answering these criticisms has relied on alteration of the WBE model itself. Enquist and collaborators account for different scaling exponents among taxonomic groups by emphasizing differences in the normalization constant _{0} of Equation 1 and deviations from the WBE assumptions regarding network geometry _{0} and with networks following exactly the geometry required by the theory. Although WBE has been frequently tested and applied

For the purpose of stating our conclusions succinctly, we refer to the “WBE framework” as an approach to explaining allometric scaling phenomena in terms of resource distribution networks (such as the vascular system) and to the “WBE model” as an instance of the WBE framework that employs particular parameters specifying geometry and (hydro)dynamics of these networks

Our main findings are: 1. The 3/4 exponent only holds exactly in the limit of organisms of infinite size. 2. For finite-sized organisms we show that the WBE model does not predict a pure power-law but rather a curvilinear relationship between the logarithm of metabolic rate and the logarithm of body mass. 3. Although WBE recognized that finite size effects would produce deviations from pure 3/4 power scaling for small mammals and that the infinite size limit constitutes an idealization

Beyond finite-size corrections we examine the original assumptions of WBE in two ways. First, we vary the predicted switch-over point above which the vascular network architecture preserves the total cross-sectional area of vessels at branchings and below which it increases the total cross-sectional area at branchings. These two regimes translate into different ratios of daughter to parent radii at vessel branch points. Second, we allow network branching ratios (i.e., the number of daughter vessels branching off a parent vessel) to differ for large and small vessels. We analyze the sensitivity of the scaling exponent with respect to each of these changes in the context of networks of finite size. This approach is similar in spirit to Price et al.

In final analysis, we are led to the seemingly incongruent conclusions that (1) many of the critiques of the WBE framework are misguided and (2) the exact (i.e., finite-size corrected) predictions of the WBE model are not fully supported by empirical data. The former means that the WBE framework remains, once properly understood, a powerful perspective for elucidating allometric scaling principles. The latter means that the WBE model must become more respectful of biological detail whereupon it may yield predictions that more closely match empirical data. Our work explores how such details can be added to the model and what effects they can have.

The paper is organized as follows. For the sake of a self-contained presentation, we start with a systematic overview of the assumptions, both explicit and implicit, underlying the WBE theory (section “Assumptions of the WBE model”). In

The WBE model rests on eight assumptions. Some of these assumptions posit the homogeneity of certain parameters throughout the resource distribution network. Any actual instance of such a network in a particular organism will presumably exhibit some heterogeneity in these parameters. The object of the theory is a network whose parameters are considered to be averages over the variation that might occur in any given biological instance. For the sake of brevity, we refer to such a network as an “averaged network”. The impact of parameter heterogenity on the scaling exponent is very difficult to determine analytically. (Section “Changing branching ratio across levels” addresses a modest version of this issue numerically.)

The relationship between metabolic rate and body mass is dominated by the structure and dynamics of the resource distribution network, which for most animals is the cardiovascular system. This assumption constitutes the core of the WBE framework. The vascular system is directly tied to metabolic rate, because the flow dynamics through the network and the number of terminal points (capillaries) constrain the rates at which cells and tissues are supplied with oxygen and nutrients needed for maintenance. At the same time, the vascular system is directly tied to body volume (and thus body mass), because network extent and structure must be such that its terminal points can service (and thus cover) the entire body volume. It follows that the relationship between metabolic rate and body mass must be constrained - and WBE assume it is dominated - by the structural and flow properties of the cardiovascular system. It should be noted that Assumption 1 could be true even if other assumptions of WBE are false. (For a recent example with plant architecture and data, see Price et al.

To say that the cardiovascular system is hierarchical amounts to assuming that there is a consistent scheme for labeling different levels of vasculature (

A vessel at level _{k}_{+1}, and lengths, _{k}_{+1}, of the two daughter vessels are identical by Assumption 3. The ratios of the radii and lengths at level _{>} and _{<} in Equations 2 and 3. The choice of _{>} for the radial ratio corresponds to area-preserving branching and of _{<} to area-increasing branching. In the WBE model, the cardiovascular system is composed of successive generations of these vascular branchings, from level 0 (the heart) to level

All the vessels at the same level of the network hierarchy have the same radius, length, and flow rate. Again, this assumption is not strictly true but provides a tractable way to study an averaged network.

The number of daughter vessels at a branching junction—the branching ratio _{k}^{k}_{k}_{−1} = ^{k}^{−1}, thus _{k}_{k}_{−1}. The constancy of

Resource distribution networks are space-filling in the sense that they must feed (though not necessarily touch) every cell in the body. This assumption determines how vessel lengths at one level relate to vessel lengths at the next level. Although this assumption seems simple and intuitively appealing, it has a precise meaning that is not easily conveyed by this terminology. A single capillary feeds a group of cells, which constitute the service volume, _{N}_{tot} = _{cap}_{N}_{cap} is the number of capillaries, that is, the number of vessels at the terminal level _{N}_{cap}. This argument can be repeated for vessels one level above the capillaries (level _{N}_{−1}. Again, the sum over all these _{N}_{−1} service volumes must equal the total volume of living tissue, _{tot} = _{N}_{−1}_{N}_{−1}, because that is the volume the capillaries must maintain. Iterating this argument over all network levels yields _{N}v_{N}_{N}_{−1}_{N}_{−1} = … = _{0}_{0}. This is the meaning of space filling. If a vessel at level _{k}_{k}_{k}_{tot}, the total volume of living tissue, is independent of

The work done to pump blood from the heart to the capillaries has been minimized by natural selection. This assumption relates the radius of vessels at one level of the network to the radius of vessels at the next level. The trial-and-error feedback implicit in evolutionary adaptation and development has led to transport networks that, on average, minimize the energy required for flow through the system. There are two independent contributions to energy loss: energy dissipated by viscous forces and energy loss due to pulse reflection at branch points. Dissipation is the major cause of energy loss in flow through smaller vessels, such as capillaries and arterioles, because a high surface-to-volume ratio subjects a larger fraction of the blood volume to friction from vessel walls. Energy loss to wave reflections, on the other hand, has the potential to be dominant in larger vessels, such as arteries, where flow is pulsatile.

Reflection can be entirely eliminated by equalizing the opposition to fluid flow before and after the branching of a vessel _{k}_{>} and _{<}) as “area-increasing” and “area-preserving” branching, respectively.

The ratio of radii in a real system probably changes continuously throughout the network. It seems, however, a reasonable approximation to assume that the ratios (3) and (4) dominate two regions, and that within each region the network is self-similar, meaning that the branching ratio _{blood}, is directly proportional to body mass

Flow rate, length, radius, hematocrit, and all other structural and physiological traits of capillaries are independent of body size. It is this assumption that allows comparisons among organisms. All previous assumptions specify the structure of the vascular system

All transfer of resources happens through the terminal exchange surfaces, i.e., at the level of capillaries and not at other levels in the network. With regard to oxygen, this assumption is well founded because capillary size and structure are likely to have been under selection pressure to facilitate the release of oxygen by red blood cells and hemoglobin

As mentioned in the introduction, it is useful to clarify some terminology that we will employ in this work. Throughout, we refer to “the WBE model” as any version based on Assumptions 1–8 above. This includes the original infinite-size limit as well as the finite-size version whose analysis we carry out in this paper. We refer to the “canonical WBE model” when singling out the predictions of the WBE model obtained with the original parameter values, such as

Using the above assumptions, we can derive how metabolic rate,

Using Assumptions 2–4 the total blood volume or total network volume (assuming the network is completely filled with blood and ignoring the factor of 2 that may arise from blood in the venous system, which returns blood to the heart) can be expressed as the sum_{<}, and _{>}, defined by Equations 2–4 resulting from Assumptions 5 and 6, to connect level _{cap} = _{N}^{N}

Given this simple relation between total blood volume (or network volume) and the number of capillaries, it is straightforward to relate metabolic rate, _{cap}. By Assumption 7 _{cap} is constant across organisms. Thus, _{cap}_{cap}, or simply _{cap}. Inserting this into Equation 10, invoking Assumption 7 that _{cap} is independent of body mass, and using Assumption 6 to recognize that _{blood} yields_{2} a constant,

Letting the number of levels in the cardiovascular system, _{cap} = ^{N}

It is essential to realize that the prediction of a 3/4 scaling relationship

We conclude that the WBE model actually predicts variation in scaling exponents due to finite-size terms whose magnitude depends on the absolute range of body masses for a given taxonomic group. These predictions can be tested against the allometric exponents reported in the empirical literature.

To quantify finite-size corrections, we focus on Equation 10 because the blood volume, _{blood} (∝_{cap} (∝_{2} and _{cap}. By inspecting Equation 10, we see that finite-size effects can become manifest in two different ways. First, even in the absence of network regions with area-increasing branching (

A network in which all levels are area-preserving corresponds to a switch-over point at _{1}|/|_{0}| = ^{−1/3}. To understand how this affects a log-log plot of V_{blood} versus the number of capillaries _{cap} (see _{cap}→∞ we obtain a scaling exponent of 3/4. However, as _{cap} decreases, the second factor in Equation 15 increases, resulting in values of _{cap} on the left of (15) that are larger than values in the case of a pure 3/4 power-law. A log-log plot of this curve will asymptote to a straight line with a slope of 3/4 for large _{cap} and bend up and away from it as _{cap} decreases. Regressing a straight line on this curve will yield a scaling exponent below 3/4, as shown schematically in

The dashed line schematically depicts the 3/4 power law that relates the number of capillaries, _{cap}, to the blood volume _{blood}. This scaling relationship is a straight line in logarithmic space (ln _{cap} versus ln _{blood}) and represents the leading-order behavior in the limit of infinite blood volume and organism size. The solid line dramatizes the curvature for the scaling relation for finite-size networks obtained when vessel radii are determined solely by area-preserving branching. The dotted line illustrates the consequences of a linear regression on the curve for finite-size organisms (solid line). Since the solid line depicts the predicted curvilinear relationship that deviates above and away from the infinite-size asymptote, Equation 16, the WBE model predicts that fits to data for organisms whose vascular networks are built only with area-preserving branching will yield scaling exponents smaller than 3/4.

We can make this quantitative by implicit differentiation, which yields the tangent to the curve defined by Equation 15 as_{cap}. Indeed, from Equation 16 we see that the tangent to the curve becomes shallower as _{cap} decreases.

In order to more directly compare these finite-size effects with empirical data, we need to develop an approach that mirrors the absolute size and size range of real taxonomic groups and organisms. To do so, we imagine constructing a group of organisms of differing sizes; the smallest organism in this group corresponds to the smallest network and the largest organism corresponds to the largest network. Each organism possesses a network with a specific value of _{cap} and _{blood} determined by Equation 10. The scaling exponent for such a group would correspond to the slope obtained from a linear regression of ln _{cap} on ln _{blood} for all of the data points obtained for all of the organisms in that group. The influence of absolute size on the scaling exponent can then be captured by fixing the size range covered by a group (e.g., 8 orders of magnitude for mammals) and measuring the change in the exponent that results from increasing the size of the organisms in the group. Consequently, if a group has a size range from the smallest to the largest organism that spans 26 levels of the vascular system, then we would compare the exponent obtained for a group covering

We now use this approach for both analytical approximations and numerical calculations. First, we can estimate analytically the exponent that would be measured for a group of organisms spanning a range of levels and thus a range of body masses. If a power-law represents a good fit for a group, we can approximate its slope using only the network (blood) volume and capillary number for the smallest and largest organisms. Hence, the slope of the regression line could be estimated by calculating the total change in ln _{cap} across the group and dividing it by the total change in ln _{blood} across the group. Using Equation 15 along with standard expansions and approximation methods, we find the leading-order terms in the limit of large _{cap} to be_{cap,S}≪_{cap,L}. The subscripts _{blood,L}/_{blood,S})≈8 ln 10. Note, however, that the numerator in Equation 17 depends on _{cap,S}, thus capturing absolute size effects, not just the range under consideration.

To test our derivations, we numerically computed and analyzed data that were generated in accordance with WBE assumptions. We start by constructing a group of different “model organisms”, each consisting of a distinct number of levels, thereby yielding a particular _{cap} and _{blood}. One might think of such a group as comprising organisms belonging to the same taxon. A group might include, for example, a smallest organism with 8 levels and a largest organism with 30 levels of vascular hierarchy. We ensured constant capillary size across all model organisms (Assumption 7) by building the networks backwards starting with the capillaries and using the scaling relationships for vessel length (2) and radius (4) conforming with Assumptions 5 and 6, respectively. Given a group of organisms so constructed, we compute the group's scaling exponent with a linear regression of ln _{cap} on ln _{blood}. An example of a regression for a particular group is shown in _{blood,L}/_{blood,S} as close as possible to the empirical value of 10^{8}. (As a guide, the number of levels in an organism varies approximately logarithmically with body mass, such that the number of levels between the largest and smallest organism is approximately _{L}_{S}_{L}_{S}^{2}>0.99) to the data within a group, yielding a group-specific scaling exponent. We then plotted the dependency of these scaling exponents on the number of capillaries in the smallest organism of each group. This protocol accounts for effects that would be observed on the basis that the smallest organism in a group (taxon) sets the “small-organism-bias” contributed by this group to the overall statistic.

(A) The logarithm of the number of capillaries is regressed with ordinary least squares (OLS) on the logarithm of blood volume for a set of artificial networks, spanning 8 orders of magnitude, built with only area-preserving branching. In this particular example the scaling exponent is determined to be 0.743, very close to 3/4. Black circles: numerical values. Red curve: power-law regression. (B) A scaling exponent _{cap,S}) in the corresponding group. Groups are built by systematically increasing the size of the smallest network, while always maintaining a range of 8 orders of magnitude in body volume (mass), resulting in the depicted graph. In all cases the branching ratio was

The results of these numerical calculations are shown in

As summarized in Assumption 6, area-increasing branching occurs not only as a consequence of minimizing dissipation in the regime of viscous flow but is also required for a portion of the network in order to match impedances for small vessels. Moreover, blood must slow down as it travels from the heart to the capillaries in order to allow for the efficient release and transfer of oxygen. By conservation of volume flow rate, the slowing of blood must be accomplished by area-increasing branching. Therefore, area-increasing branching has a significant influence even on the scaling exponent of groups dominated by large organisms. This influence only increases in groups biased towards smaller organisms in which a large fraction of the network exhibits area-increasing branching to minimize dissipation.

Here we analyze the limiting case of a network in which all levels are area increasing. This corresponds to a transition at _{0} and _{1} are no longer constant with respect to _{blood}). However, returning to Equation 6 we note that only the second sum survives (_{cap} = ^{N}_{cap}/ln _{blood}) ranges are considered. As is evident from the expression for the tangent to the curve defined by Equation 18,

In complete analogy to section “Networks with only area-preserving branching”, we estimate the scaling exponent that would be measured for groups of organisms spanning a range of levels using only the difference in the logarithms of network volume and capillary number between the smallest and largest specimens. The leading-order expression is given by_{S}_{L}_{S}_{L}_{S}_{L}

As before, we constructed artificial datasets in accordance with WBE assumptions, but using the area-increasing relationship for vessel radii, Equation 3. The results of these numerical calculations are shown in _{S}_{L}

As in _{S}_{L}_{S}_{L}_{S}_{L}

We conclude that finite-size effects on the scaling exponent are much more important for networks entirely composed of area-increasing branching than for networks operating entirely in the regime of area-preserving branching, described in section “Networks with only area-preserving branching”. The differential impact of finite-size effects in the two extreme cases is crucial for understanding finite-size effects in mixed networks with a transition between the two branching regimes.

The original WBE theory assumes that the cardiovascular system is a combination of area-preserving and area-increasing regimes. In large vessels, blood flow is predominantly pulsatile and the pulse wave can lose energy through reflections at vessel branch points. Minimizing this type of energy loss leads to the requirement that the total cross-sectional area of daughter vessels must preserve the area of the parent vessel in the early part of the network and switch to area-increasing branching farther downstream. In small vessels, on the other hand, blood flow is viscous. When optimizing energy expenditure for the viscous transport of blood, dissipation due to frictional drag from vessel walls becomes important. Minimizing such dissipation requires area-increasing branching, as summarized in Assumption 6, Equation 3. Optimizing these two flow regimes leads to a transition from area-preserving to area-increasing branching. WBE calculate the transition level to be

In the WBE model, these values for the transition level also set the size of the smallest organism, a mammal in which a heart beat cannot be sustained because the vessels are so small that the pulsatile flow is immediately dissipated. This is suggested in WBE by the assertion that “In a 3 g shrew, Poiseuille flow begins to dominate shortly beyond the aorta” _{S}_{S}

The full form of Equation 10 describes this case. Recognizing that

In comparison to Equation 15 the sign of _{1} has changed, and this reverses, with significant consequences, our previous arguments. Specifically, as _{cap} decreases, _{cap} on the left of (21) than the values predicted from a pure 3/4 power-law. A log-log plot of this curve will asymptote to a straight line with a slope of 3/4 for large _{cap} and will curve down and away from this asymptote as _{cap} decreases. This effect derives from the fact that small mammals exhibit scaling exponents >3/4, a point raised by WBE in their original work _{cap} decreases:_{cap}.

The dashed line schematically depicts the 3/4 power law of ln _{cap} versus ln _{blood} in the infinite network limit. The solid line dramatizes the curvature for the scaling relation that is obtained when the network has a transition point above which it has area-preserving branching and below which it has area-increasing branching. The dotted line illustrates the consequences of a linear regression on what is a curvilinear relationship that deviates below and away from the infinite-size limit, Equation 22. As a result, the WBE model predicts that fits to data for organisms whose vascular networks are built in mixed mode will yield scaling exponents that are larger than 3/4.

Using the same estimation procedure as in previous sections, we find that the scaling exponent computed over a given range of _{cap}_{,S}, which is tied to the number of levels in the smallest organism, captures absolute size effects. It makes the scaling exponent sensitive to the contributions from small organisms when groups span a range above their smallest member.

We generated artificial data as in the previous two sections, including a transition between area-preserving and area-increasing ratios of vessel radii, Equations 4 and 3, at

In

(A) As in _{cap,S}). Many groups are built by systematically increasing the size of the smallest network, resulting in the depicted graph. In all cases the branching ratio was _{S}_{S}_{S}

As anticipated, the branching ratio has virtually no effect on the scaling exponent predicted by the WBE model. This is both because the leading-order term of 3/4 does not depend on ^{(N̅−1)/3}^{−(N̅+1)/3} = ^{−2/3}. Hence, the first-order correction becomes ^{−2/3}^{1/3}−1)/ln(_{blood,L}/_{bllod,S}), evidencing a very weak dependence on the branching ratio _{blood,L}/_{blood,S}) or _{blood,L}/_{blood,S}) = ln(10^{8})≈18.4 _{0} the pulse wave velocity according to the Korteweg-Moens equation, and _{cap} and _{cap} are the mean radius and length of a capillary, respectively

The results of this section suggest that a strict test of the canonical WBE model should compare measured exponents to 0.81 rather than 3/4. Alternatively, one might argue that for the WBE model to yield a 3/4 exponent, the cardiovascular system of the smallest mammal must comprise many more than

We find that finite-size effects change measured scaling exponents for networks with pure area preserving, pure area increasing or a mixture of both. Empirically determined scaling exponents for basal metabolic rate are typically a little lower than

The exponent of

The discrepancies between the WBE model and data might be addressed in several ways: (i) by correcting for biases in the empirical distributions of species masses; (ii) by adding more detail to any of the WBE assumptions; (iii) by relaxing the assumptions. In

The WBE approach determines the transition level from area-preserving to area-increasing branching by equating the impedance in regions with pulsatile flow with the impedance (resistance) in regions of smooth (Poisueille) flow

There are reasons to doubt the assumptions behind WBE's calculation of the number of levels with area-increasing branching,

Addressing these problems will eventually require detailed hydrodynamical calculations and extensive knowledge of the cardiovascular system. We can, however, illustrate the effect of a change in _{S}_{blood} that is 8 orders of magnitude larger than that of the smallest organism. These results are summarized in

The scaling exponent

We also examined the consequences of a transition region, rather than a single transition level, from area-preserving to area-increasing branching. We spread the transition out over

The three curves are analogous to those in _{k}_{+1}/_{k}^{−1/2} = 0.707 to ^{−1/3} = 0.794 over 12 levels centered at the WBE transition level

All calculations thus far assume that the branching ratio _{b}_{b}_{b}_{b}_{0} the pulse wave velocity according to the Korteweg-Moens equation (for definitions see, for example, _{trans}≈1 mm, using the same numbers as WBE, that is, ^{2}, _{cap} = 0.08 mm, ^{3}, _{0} = 600 cm/s, _{cap} = 4 µm. As argued above, there are reasons to doubt this calculation for the radial scaling transition. We employ the same formula, however, since that allows us to make a direct comparison with the canonical WBE model and lets us consider just the effects of changing the branching ratio.

As before, networks for the changing-_{b}_{trans}, branching changed from area-increasing to area-preserving. We held constant the number of levels, _{b}_{blood}.

As shown in

The scaling exponent _{b}_{b}

Savage et al.

The data are binned in orders of magnitude for body mass as described in the text. (A) Cumulative binning starting with smallest mammals. (B) Cumulative binning starting from largest mammals. (C) Exponents from individual order-of-magnitude bins. The exponents computed from these aggregations of empirical data vary both above and below 3/4. Note, however, that in all cases the allometric exponents tend to increase with increasing body mass. The error bars represent the 95% confidence intervals. When data is scarce, the confidence intervals become so large that the exponents cannot be trusted. (The full range of some error bars is cut off by the scale of the plots.)

The panels of

The concave increase of the scaling exponent with body mass is most consistent with a finite-size WBE model based on pure area-preserving branching throughout the network, see section “Networks with only area-preserving branching”. (The concave increase of the scaling exponent,

A similar analysis of a more limited dataset for heart rate (26 data points) and respiratory rate (22 data points) ^{−α/3} (see Table S1 and related text in section “Impact of finite-size corrections on additional WBE predictions” of ^{−0.27} and asymptote to −1/4 with increasing mass. That is, there should be very little change in the scaling exponent when analyzing data for either small or large mammals. This does not match empirical heart rate data well. Regressing on the first three, four, and six orders of magnitude in body mass yields exponents of −0.33, −0.27, and −0.25, respectively. The match is worse for respiratory rate data. Regressing on the first two, three, five, and seven orders of magnitude in body mass gives exponents of −0.64, −0.44, −0.34, and −0.26, respectively. We observe a convergence to −1/4, but over a much larger range of scaling exponents than expected.

While the WBE model has been predominantly interpreted in the context of interspecific scaling

It is important to note that empirical data for the inter- and intraspecific case (especially for restricted size classes) are rather limited. We therefore do not wish to overstate the strength of our conclusions. We merely report discrepancies between the predictions of the canonical WBE model and limited sets of data. We anticipate that further data acquisition, statistical analysis, and model refinement will bring theory and data into agreement.

Over the past decade, the WBE model has initiated a paradigm shift in allometric scaling that has led to new applications (e.g.,

In section “Assumptions of the WBE model”, we provide a detailed presentation of the complete set of assumptions and calculations defining the WBE model. While none of these originated with us, the literature lacked, surprisingly, an exhaustive exposition. (In particular, the consequences of Assumption 6 are a distillation of hydrodynamical calculations that we summarize in

One of our main objectives is to clarify that the WBE model predicts (and thus “explains”) the 3/4 exponent of the scaling law relating whole-organism metabolic rate to body mass

A major consequence of the curvilinear relationship between ln

Furthermore, we find evidence for size-dependent relationships in the available empirical data for mammals (section “Comparison to empirical data”). Specifically, we find that the measured scaling exponent tends to _{cap} versus ln _{blood}) exhibits convex curvature (i.e., the type of relationship dramatized in

The case for pure area-increasing branching (hypothesis (i) above) within the WBE model is somewhat problematic. The only way for such a network to be consistent with Assumption 6 would be to posit that the transition from area-preserving to area-increasing regimes occurs at a vessel radius ^{−1/3}, strongly implying that area-increasing branching is in fact dominant when vessel radii are small

In our hands, empirical data seem most consistent with networks built with purely area-preserving branching, although the lack of very high-quality data for both metabolic rate and body mass makes it difficult to be absolutely certain of this trend. The reasoning outlined above makes hypothesis (i) appear somewhat unlikely. This leaves us with a riddle: cardiovascular networks with architectures that support the scaling trends observed for real organisms would seem to violate Assumption 6 of the WBE model. We are thus led to believe that some modification of assumptions 2–8 is necessary to explain the concavity in the data and an empirical scaling exponent less than 3/4. While a model that aligns with the empirical evidence might differ from the canonical WBE model (assumptions 2–8 plus specific values for the parameters

Resolving this paradox will likely require intensive further data analysis and extension of the canonical WBE model. It is clear that work in this area would benefit from a more detailed empirical understanding of cardiovascular networks themselves. Although data for the coronary artery in humans, rats, and pigs exist _{k}_{+1}/_{k}_{k}_{+1}/_{k}_{k}_{+1}/_{k}

In this paper we have begun the process of relaxing some assumptions of the canonical model. Although these modifications produce interesting results, they do not fully address the riddles discussed above. Addition of further biological realism, such as asymmetric branching or the flow characteristics of the slurry of blood cells at small vessel sizes, may generalize the WBE model from an asymptotic predictor of metabolic scaling into a universal theory that provides an understanding of which properties of resource distribution networks are most relevant for metabolic scaling in any given biological context. This will enable testing the very soundness of the WBE framework (Assumption 1) and the extent to which the cardiovascular system shapes one of the most wide ranging regularities across animal diversity.

Minimizing energy loss to dissipation. Plot for an arbitrary level of the cardiovascular network (^{k}r^{4})+^{k}r^{2}^{2k}^{6} = 1/512≈0.00195. We plot the first two terms of the objective function versus

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Minimizing energy loss to reflection. Plot of the reflection coefficient squared (|^{2} = |(1−_{k}_{k}_{+1})/(1+_{k}_{k}_{+1})|^{2}) versus the ratio of vessel radii _{k}_{+1}/_{k}^{−6} s/m^{2}. We choose a bifurcating branching ratio ^{−1}/^{2}, which exactly corresponds to area-preserving branching and impedance matching.

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Minimizing total power loss. The graph depicts the _{k}_{k}_{+1}/_{k}_{k}_{cap}^{−6} s/m^{2} and for wave frequency of _{0} = 1.17 s^{−1}. For smaller values of _{k}_{k}_{k}_{k}_{k}^{−1/3} = 2^{−1/3} = 0.794 and _{k}^{−1/2} = 2^{−1/2} = 0.707 extremely well. Also, note that the transition from area-preserving to area-increasing branching begins at _{k}_{k}

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Log-normal sampling bias for small networks. A single realization of 1000 numerically generated data points for networks with a branching ratio of

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Sizing Up Allometric Scaling Theory

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The authors are particulary grateful to Geoffrey West for many insightful discussions and acknowledge being profoundly influenced by his style of thought in framing the biological scaling problem. The authors benefited from many interactions with Jim Brown and Brian Enquist, fountains of biological knowledge in the field. Thomas Kolokotrones and Daniel Yamins provided good reasoning in statistics and helped with conceptual hygiene. Brian Enquist and Scott Stark gave the manuscript a thorough and rigorous reading, which led to several improvements. We gratefully acknowledge five referees for their time and constructive criticism above and beyond the collegial call of duty. VMS wishes to acknowledge Karen Christensen-Dalsgaard for encouraging him to consider the importance of finite-size corrections to the WBE scaling theory.

_{O2,max}and aerobic scope in mammals.